Lattice QCD results for mesons containing b quarks from the HPQCD collaboration CONFINEMENT XISt Petersburg

Outline  Radiative improvement of NRQCD using background field approach.  Lattice action and most recent configurations.  Quarkonium and -meson spectrum; hyperfine splittings.  B-meson decay constants:.  mixing.  Mass of the b quark.  decay  The future for heavy quarks: NRQCD and HISQ.

Radiatively Improved NRQCD Evolve heavy quark Green’s function with kernel: where At tree level Radiatively improve to 1-loop using Background Field Method Also include certain four-fermion operators in NRQCD action:

Matching with Background Field Method NRQCD is an effective theory containing operators with D > 4 which, at tree level, can be restricted to be gauge-covariant. Vital to use formulation where no non-covariant operators are generated by radiative processes. Gauge invariance is retained by the method of background field gauge, and ensures gauge invariance of the effective action. All counter-terms are FINITE in BFG => can compute ALL matching, both continuum relativistic and non-relativistic, using lattice regularization: QED-like Ward Identities. Derive 1PI gauge-invariant effective potential. Match (on-shell) S-matrix. Implemented in HiPPY and HPsrc for automated lattice perturbation theory.

Matching with Background Field Method In particular, evaluate the spin-dependent diagrams vital for accurate evaluation of hyperfine structure: Improve also: Currents for decays Wilson operators for mixing

Most Recent Configurations HISQ staggered (u,d,s,c sea quark) configurations generated by the MILC collaboration with radiatively improved gluon action. Lattice spacing generally determined by splitting. Errors are statistics, NRQCD systematics, experiment. Sets 3,6,8 are at the physical point; no chiral extrapolation necessary. Results presented use these and also MILC 2+1 Asqtad configurations.

The gold-plated spectrum - HPQCD Use 2 nd generation, HISQ sea: sets 1,2,4,5,7 (not at physical point) Radiatively improved NRQCD for b quark with HISQ u,s,d,c valence quarks.

Hyperfine splittings: Already, so improvement to spin-dependent NRQCD operators vital Set 1 Set 2 Set 4 Set 5 Set 7

Decay Constants B-meson decay constants, : First LQCD results for with physical light quarks: LQCD predicts: Experiment within 1-s.d. of lattice. Error mainly experimental, but also some uncertainty in.

Using world-average, HPQCD, results for find crucial to decay, and hitherto major source of error: Second error, from, now competitive. electromagnetic decay constant: is n-th term in derivative expansion of the NRQCD lattice current. determined non-perturbatively using JJ correlator.

Decay constant summary plot

mixing First LQCD calculation of mixing parameters with physical light quark masses at three lattice spacings: MILC HISQ sets 3,6,8 Compute matrix elements of effective 4-quark operators derived from box diagrams using LQCD: needed for oscillations All three appear in width difference Use radiatively improved NRQCD Bag parameters defined by Similarly for with, respectively

Operators: NRQCD b-quark and HISQ light quark matched to continuum: Results preliminary: more accurate results at physical point imminent. Using expt., current HPQCD analysis gives: (PDG: )

b-quark mass known to 3-loop order is the energy of the meson at rest using NRQCD on the lattice is computed fully to 2-loops in perturbation theory as follows: 1.Measure on high- quenched gluon configurations using heavy quark propagator in Landau gauge with t’Hooft twisted boundary conditions. 2. Fit to 3 rd -order series in and extract quenched 2-loop coefficient. 3. Compute 2-loop contribution using automated perturbation theory for b-quark self-energy at p = 0. Similar theory using meson instead.

Fit to consistent with known 1-loop automated pert. th. result Include 3-loop quenched coefficient in. Error dominated by 3-loop contribution. for two lattice spacings using both Most accurate to use and correlator method. Applicable for HISQ and NRQCD valence quarks. For HISQ valence need extrapolation in some cases from

since is not renormalized. RHS from 3-loop calculation of Chetyrkin et al. LHS from LQCD. Use moments. Fit to extract. For this method gives Weighted lattice average for measurements on configs with 3,4 sea quarks and then run to :

and Highly suppressed in Standard Model: test for BSM physics Recently measurements by LHCb and data also from CDF, ATLAS, CMS Describe observables by non-perturbative form factors. Calculate form-factors at large with NRQCD on lattice. Complements results of other analyses including sum rules at low BSM physics Relevant operators: Look for departure from SM  BSM physics Wilson coeffs (RRH with Z. Liu, S. Meinel, M. Wingate: , )

Example of tension between SM and data Shaded bands show theory errors. Average over bins conforming to data bins. Best fit allowing BSM coefficients : An indication that BSM physics contributes? Agrees with other analyses: , , , Caution: signal intriguing but not yet properly significant. Caution: need analysis of charmonium resonance effects

The Future: NRQCD and HISQ Selected quantities: NRQCD: Have radiatively improved coefficents and operators. HISQ: Fully relativistic and applicable for. Next generation configs: physical sea quarks; incorporate QED effects. Improve QCD parameters:, quark masses and hadronic matrix elements. See for relevance of accurate, h=c,b, to higgs physics and future collider programmes.

C. Davies, J. Koponen, B. Chakraborty, B. Colquhoun, G. Donald, B. Galloway (Glasgow) G.P. Lepage (Cornell) G. von Hippel (Mainz) C. Monahan (William and Mary) A.Hart (Edinburgh) C. McNeile (Plymouth) RRH, R. Dowdall, T. Hammant, A. Lee (Cambridge) J. Shigemitsu (Ohio State) K. Hornbostel (Southern Methodist Univ.) H. Trottier (Simon Fraser) E. Follana (Zaragoza) E. Gamiz (CAFPE, Granada) HPQCD: recent past and present